Some Results on the Extended Hypergeometric Matrix Functions and Related Functions

The special matrix functions have received significant attention in many fields, such as theoretical physics, number theory, probability theory, and the theory of group representations. In 2017, Dwivedi and Sahai introduced the generalized hypergeometric matrix function using matrix parameters and proved the convergence on | z | = 1 {|z|=1} . Recently, hypergeometric matrix functions and their potential applications have played a major role in mathematical physics and engineering. Motivated by aforesaid works and in order to enrich this flourishing field, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and determine various transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.

The present paper deals with the definition and study of new kinds of hypergeometric matrix functions within complex analysis. We also get the radius of regularity and matrix recurrence relations are then developed for l(m,n)-hypergeometric matrix function of two complex variables. We give a different approach to prove the effect of the differential operator on this function. Finally, another hypergeometric matrix function, namely, p l(m,n)-hypergeometric matrix function of two complex variables are defined, its components when the positive integer p is greater than one, provide a matrix partial differential equation satisfied by these function and some of their properties are investigated.

Many authors defined and extended the beta function in various forms because the beta function has wide uses in different fields of science and applied science. In this article, we define a new more generalized form of the extended beta matrix function via the Wiman matrix function and describe their significant properties and special cases. Furthermore, we define an extension of the Gauss hypergeometric and confluent hypergeometric matrix functions by adopting a novel type of beta matrix function. We also derive their Laplace transform, derivative formula and transformation formulae.

In this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

Dirichlet series associated with the Poincaré series attached to mathrm{SL}(2,{{mathbb {Z}}}) are introduced. Integral representations and transformation formulas are given, which describe the Voronoï-type summation formula for the exponential-type generating function of the Riemann zeta-function. As an application, a new proof of the Fourier series expansion of holomorphic Poincaré series is given.

An extension of the Beta matrix function by introducing an extra matrix parameter is applied here to extend the Gauss and Kummer hypergeometric matrix functions. Since the latter matrix functions contain several of the familiar special matrix functions as sub-cases. It is expected that our extension is useful. Moreover, some properties of the extended Gauss and Kummer hypergeometric matrix functions are derived.

Reliability, survivability, as well as the optimal operating mode of operation of the supercomputer will depend on the architecture and efficiency of the cooling system of the hot components of the supercomputer. That is why the number of problems, of great theoretical and practical interest, is the problem of studying the temperature fields arising in elements of arbitrary configuration, cooling a supercomputer. To solve this class of heat conduction problems, the method of finite integral transformations turned out to be the most convenient. This article is the first to construct a new finite integral transformation for the Laplace equation in an arbitrary domain bounded by several closed piecewise-smooth contours. An inverse transformation formula is given. Finding the core of the constructed new finite integral transformation by the finite element method in the Galerkin form for simplex first-order elements reduces to solving a system of algebraic equations. To test the operability of the new integral transformation, calculations were carried out of solutions of the boundary value problem for the Laplace equation obtained using the developed new integral transformation and the well-known analytical solution. The results of comparison the calculations of the solution of the Laplace equation are presented. In the case of a square with a side length equal to one and on one side of the square, the temperature is unity, and on the other, the temperature is zero, with a well-known analytical solution and a solution obtained using the new integral transformation. These results were obtained for 228 simplex first-order elements and 135 nodes. The maximum deviation modulo of these solutions is 0,096, the mathematical expectation of deviations is 0,009, and the variance of the type is 0,001. The developed integral transformation makes it possible to obtain a solution to complex boundary value problems of mathematical physics.

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.

Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations.

view Abstract Citations (12) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The calculation of oscillator strengths and multipole radial integrals in the Coulomb approximation. Binh, D. H. ; Prudhomme, M. ; van Regemorter, H. Abstract A very simple method is presented which yields accurate values of multipole radial integrals in the Coulomb approximation, even for transitions involving high Rydberg states. It is noted that the smallest of the effective quantum numbers can be as large as 200. The previously described approach is briefly outlined with a slightly more general formulation applicable to any multipole transition in any ion with a specified net charge. A recurrence relation is pointed out which makes the calculation of the Gordon (1929) formula quite rapid and easy. Publication: The Astrophysical Journal Pub Date: June 1979 DOI: 10.1086/182976 Bibcode: 1979ApJ...230L.127B Keywords: Coulomb Potential; Integral Transformations; Multipoles; Oscillators; Rydberg Series; Transition Probabilities; Astrophysics; Atomic Spectra; Quantum Numbers; Atomic and Molecular Physics; Oscillator Strengths full text sources ADS |

Integral transformations of the form from Lp(ℝ+) into Lq(ℝ+); (1 ≤ p ≤ 2, p−1 + q −1 = 1) are studied. Necessary and sufficient conditions on k to ensure that the transformation is unitary on L2(ℝ+) are obtained, and a formula for the inverse transformation is derived in this case.

The aim of this paper is to extend the domain of polygamma function from the set of complex numbers to the set of bicomplex numbers. We also discuss integral representation, recurrence relation, multiplication formula and reflection formula for this function.

Using expansion formula for the translation of Slater-type orbitals (STOs) obtained with the help of Coulomb Sturmian exponential-type functions (ETFs) (Guseinov, I. I. Int J Quantum Chem 2001, 81, 126) the multicenter electron-repulsion integrals are expressed in terms of the basic overlap integrals with the same screening parameters and the two-center basic hybrid integrals. For the calculation of basic hybrid integrals the recurrence relations are derived. These recurrence relations enable us to calculate some basic hybrid integrals by the use of which other basic integrals are calculated analytically from the formulas in terms of new auxiliary functions introduced recently by the authors. The recurrence relations obtained in this work for basic hybrid integrals are especially useful for the calculation of multicenter electron-repulsion integrals for large quantum numbers of STOs.

Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.

An integral transform $\bar f(s) = \int f(t)\varphi (s,t)dt$ can be averaged over the convex hull of $\{ s_1 , \cdots ,s_k \} $ to produce an analogous function $\bar F\{ s_1 , \cdots ,s_k \} $ of several real or complex variables. The question arises whether it is legitimate to take the average under the integral sign, so that $\bar F\{ s_1 , \cdots ,s_k \} = \int f(t)\Phi (s_1 , \cdots ,s_k ;t)dt$, where $\Phi $ is the corresponding average of $\varphi $. Conditions for the validity of this equation are of interest in the theory of special functions because the kernel $\Phi $ may be a Bessel function, elliptic integral, or other hypergeometric function when $\varphi $ is the kernel of a Laplace, Fourier, or Stieltjes transformation. Sufficient conditions of validity are established in the case of these three transformations and the inverse Laplace, Fourier, and Mellin transformations. The averages have some but not all of the operational properties of the corresponding ordinary transforms. Some examples are given involving various special functions.